We construct such a spread distribution on matchings via iterative absorption, which was first introduced by Kühn and Osthus [1]} and Knox, Kühn, and Osthus [2]} to prove results on Hamilton decompositions. This powerful method has played a prominent role in several recent breakthroughs. We mention only a proof that combinatorial designs exist [3]} and the proof of the Erdős–Faber–Lovász conjecture [4]}. The method is substantially simplified in the context of perfect matchings.
